$\ast$-Ricci-Yamabe Soliton and Contact Geometry

TL;DR

This paper introduces and studies the concept of $ extasterisk$-Ricci-Yamabe solitons on contact metric manifolds, classifies non-Sasakian cases, and characterizes conditions under which Sasakian 3-manifolds exhibit specific geometric properties.

Contribution

It extends the notion of $ extasterisk$-Ricci solitons to $ extasterisk$-Ricci-Yamabe solitons on contact manifolds and classifies non-Sasakian cases, providing new insights into their geometry.

Findings

Classification of non-Sasakian $N(k)$-contact metric manifolds with $ extasterisk$-Ricci-Yamabe solitons.

Sasakian 3-manifolds admitting such solitons are $ extasterisk$-Ricci flat and have Fano transverse geometry.

Sasakian 3-metrics are homothetic to Berger spheres and the solitons are steady.

Abstract

It is well known that a unit sphere admits Sasakian 3-structure. Also, Sasakian manifolds are locally isometric to a unit sphere under several curvature and critical conditions. So, a natural question is: Does there exist any curvature or critical condition under which a Sasakian 3-manifold represents a geometrical object other than the unit sphere? In this regard, as an extension of the $\ast$-Ricci soliton, the notion of $\ast$-Ricci-Yamabe soliton is introduced and studied on two classes contact metric manifolds. A $(2n + 1)$-dimensional non-Sasakian $N(k)$-contact metric manifold admitting $\ast$-Ricci-Yamabe soliton is completely classified. Further, it is proved that if a Sasakian 3-manifold $M$ admits $\ast$-Ricci-Yamabe soliton $(g,V,\lambda,\alpha,\beta)$ under certain conditions on the soliton vector field $V$, then $M$ is $\ast$-Ricci flat, positive Sasakian and the transverse geometry of $M$ is Fano. In addition, the Sasakian 3-metric $g$ is homothetic to a Berger sphere and the soliton is steady. Also, the potential vector field $V$ is an infinitesimal automorphism of the contact metric structure.